Some vanishing results for the rational completed cohomology of Shimura varieties

TL;DR

This paper proves that in the cohomology of Shimura varieties, only middle degree cohomology contains sufficiently regular infinitesimal weights, based on a vanishing result in mixed characteristics.

Contribution

It establishes a vanishing theorem for certain weights in the cohomology of Shimura varieties, extending previous results to a more general setting.

Findings

Sufficiently regular weights appear only in the middle degree cohomology.

The proof relies on an almost Kodaira-type vanishing result in mixed characteristics.

The result applies to the locally analytic completed cohomology of Shimura varieties.

Abstract

Based on an almost Kodaira-type vanishing result in mixed characteristics of Bhatt, we show that, in the locally analytic completed cohomology of a general Shimura variety, sufficiently regular infinitesimal weights can only show up in the middle degree.